Hecatonicosihexacositruncated hecatonicosihexacosichoron

Hecatonicosihexacositruncated hecatonicosihexacosichoron
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Hixthix
Coxeter diagram	o3x3x5/3x5*b ()
Elements
Cells	600 truncated tetrahedra, 120 truncated dodecahedra, 120 icosidodecatruncated icosidodecahedra
Faces	2400 triangles, 2400 hexagons, 1440 decagons, 720 decagrams
Edges	3600+3600+7200
Vertices	7200
Vertex figure	Sphenoid, edge lengths 1 (1), √3 (2), √(5+√5)/2 (2), and √(5–√5)/2 (1)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {43+15{\sqrt {5}}}{2}}}\approx 6.18632$
Hypervolume	$25{\frac {2193+1192{\sqrt {5}}}{4}}\approx 30364.9564$
Dichoral angles	Idtid–6–tut: $\arccos \left(-{\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 157.76124^{\circ }$
	Idtid–10/3–idtid: 144°
	Idtid–10–tid: 144°
	Tut–3–tid: $\arccos \left(-{\frac {\sqrt {10}}{4}}\right)\approx 142.23876^{\circ }$
Number of external pieces	5520
Level of complexity	36
Related polytopes
Army	Semi-uniform Grahi, edge lengths ${\frac {3-{\sqrt {5}}}{2}}$ (decagons), 1 (triangles)
Regiment	Hixthix
Conjugate	Hecatonicosihexacosiquasitruncated hecatonicosihexacosichoron
Convex core	Hecatonicosachoron
Abstract & topological properties
Flag count	172800
Euler characteristic	–1080
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

The hecatonicosihexacositruncated hecatonicosihexacosichoron, or hixthix, is a nonconvex uniform polychoron that consists of 600 truncated tetrahedra, 120 truncated dodecahedra, and 120 icosidodecatruncated icosidodecahedra. 1 truncated tetrahedron, 1 truncated dodecahedron, and 2 icosidodecatruncated icosidodecahedra join at each vertex.

Cross-sections

Vertex coordinates

The vertices of a hecatonicosihexacositruncated hecatonicosihexacosichoron of edge length 1 are given by all even permutations of:

• $\left(0,\,\pm {\frac {1}{2}},\,\pm {\frac {13+3{\sqrt {5}}}{4}},\,\pm {\frac {3+2{\sqrt {5}}}{2}}\right),$
• $\left(0,\,\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {11+5{\sqrt {5}}}{4}},\,\pm {\frac {1+2{\sqrt {5}}}{2}}\right),$
• $\left(0,\,\pm {\frac {1}{2}},\,\pm 3{\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}}\right),$
• $\left(0,\,\pm {\frac {1}{2}},\,\pm {\frac {9+5{\sqrt {5}}}{4}},\,\pm {\frac {3+5{\sqrt {5}}}{4}}\right),$
• $\left(0,\,\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm (3+{\sqrt {5}}),\,\pm (1+{\sqrt {5}})\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+3{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{2}},\,\pm {\frac {3}{2}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {7+5{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {4+{\sqrt {5}}}{2}},\,\pm 3{\frac {3+|sqrt5}{4}},\,\pm {\frac {5+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm {\frac {11+3{\sqrt {5}}}{4}},\,\pm (2+{\sqrt {5}})\right),$
• $\left(\pm {\frac {1}{2}},\,\pm 1,\,\pm {\frac {7+5{\sqrt {5}}}{4}},\,\pm 5{\frac {1+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3{\sqrt {5}}-1}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm 3{\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {3{\sqrt {5}}-1}{4}},\,\pm (3+{\sqrt {5}}),\,\pm {\frac {5+3{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {7+{\sqrt {5}}}{4}},\,\pm {\frac {11+3{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm 5{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm 1,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm (2+{\sqrt {5}}),\,\pm {\frac {5+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {7+5{\sqrt {5}}}{4}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {11+5{\sqrt {5}}}{4}},\,\pm {\frac {7+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {9+5{\sqrt {5}}}{4}},\,\pm {\frac {4+{\sqrt {5}}}{2}}\right),$
• $\left(\pm 1,\,\pm {\frac {3{\sqrt {5}}-1}{4}},\,\pm {\frac {11+5{\sqrt {5}}}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right),$
• $\left(\pm 1,\,\pm {\frac {3}{2}},\,\pm {\frac {11+3{\sqrt {5}}}{4}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}}\right),$
• $\left(\pm 1,\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {13+3{\sqrt {5}}}{4}},\,\pm {\frac {4+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3}{2}},\,\pm (2+{\sqrt {5}}),\,\pm 5{\frac {1+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm 3{\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+3{\sqrt {5}}}{4}},\,\pm {\frac {13+3{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3{\sqrt {5}}-1}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm (2+{\sqrt {5}}),\,\pm {\frac {4+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {3+5{\sqrt {5}}}{4}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {4+{\sqrt {5}}}{2}},\,\pm (1+{\sqrt {5}}),\,\pm 5{\frac {1+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3{\sqrt {5}}-1}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm (3+{\sqrt {5}}),\,\pm {\frac {7+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3}{2}},\,\pm {\frac {1+3{\sqrt {5}}}{4}},\,\pm {\frac {9+5{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {7+{\sqrt {5}}}{4}},\,\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm 5{\frac {1+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {1+2{\sqrt {5}}}{2}},\,\pm {\frac {11+3{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {3+5{\sqrt {5}}}{4}},\,\pm (2+{\sqrt {5}})\right),$
• $\left(\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {1+2{\sqrt {5}}}{2}},\,\pm {\frac {7+5{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {1+3{\sqrt {5}}}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm (1+{\sqrt {5}}),\,\pm {\frac {11+3{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {1+3{\sqrt {5}}}{4}},\,\pm {\frac {7+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {7+5{\sqrt {5}}}{4}}\right).$

External links

• Bowers, Jonathan. "Category 8: Great Rhombates" (#325).

• Klitzing, Richard. "hixthix".

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